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Chapter 6 Exponential and Logarithmic Functions

6.3 Logarithmic Functions

Learning Objectives

In this section, you will:

  • Convert from logarithmic to exponential form.
  • Convert from exponential to logarithmic form.
  • Evaluate logarithms.
  • Use common logarithms.
  • Use natural logarithms.
Photo of the aftermath of the earthquake in Japan with a focus on the Japanese flag.
Figure 1 Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)

In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes.4 One year later, another stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,5 like those shown in Figure 1. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale,6 whereas the Japanese earthquake registered a 9.0.7

The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is 1084=104=10,000 times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.

Converting from Logarithmic to Exponential Form

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is10x=500, wherexrepresents the difference in magnitudes on the Richter Scale. How would we solve forx?

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve10x=500.We know that102=100and103=1000, so it is clear thatxmust be some value between 2 and 3, sincey=10xis increasing. We can examine a graph, as in Figure 2, to better estimate the solution.

 

Graph of the intersections of the equations y=10^x and y=500.
Figure 2

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 2 passes the horizontal line test. The exponential functiony=bxis one-to-one, so its inverse,x=byis also a function. As is the case with all inverse functions, we simply interchangexandyand solve foryto find the inverse function. To representyas a function ofx, we use a logarithmic function of the formy=logb(x).The baseblogarithm of a number is the exponent by which we must raisebto get that number.

We read a logarithmic expression as “The logarithm with basebofxis equal toy” or, simplified, “log basebofxisy.” We can also say, “braised to the power ofyisx,” because logs are exponents. For example, the base 2 logarithm of 32 is 5 because 5 is the exponent we must apply to 2 to get 32. Since25=32, we can writelog232=5.We read this as “log base 2 of 32 is 5.”

We can express the relationship between logarithmic form and its corresponding exponential form as follows:

logb(x)=yby=x,b>0,b1

Note that the basebis always positive.

Think b to the y equals x.

Because logarithm is a function, it is most correctly written aslogb(x), using parentheses to denote function evaluation, just as we would withf(x).However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, aslogbx.Note that many calculators require parentheses around thex.

We can illustrate the notation of logarithms as follows:

logb (c) = a means b to the A power equals C.

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This meansy=logb(x)andy=bxare inverse functions.

Definition of the Logarithmic Function

A logarithm basebof a positive numberxsatisfies the following definition.

Forx>0,b>0,b1, y=logb(x) is equivalent to by=x where,

  • we readlogb(x)as, “the logarithm with basebofx” or the “log basebofx.
  • the logarithmyis the exponent to whichbmust be raised to getx.

Also, since the logarithmic and exponential functions switch thexandyvalues, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

  • the domain of the logarithm function with base b is (0,).
  • the range of the logarithm function with base b is (,).

 

Can we take the logarithm of a negative number?

No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.

 

How To

Given an equation in logarithmic formlogb(x)=y, convert it to exponential form.

  1. Examine the equationy=logbx and identifyb,y,andx.
  2. Rewritelogbx=yasby=x.

Converting from Logarithmic Form to Exponential Form

Write the following logarithmic equations in exponential form.

  1. log6(6)=12
  2. log3(9)=2
Show Solution

First, identify the values ofb,y,and x.Then, write the equation in the formby=x.

  1. log6(6)=12Here,b=6,y=12,and x=6.Therefore, the equationlog6(6)=12is equivalent to612=6.
  2. log3(9)=2 Here,b=3,y=2,and x=9.Therefore, the equationlog3(9)=2is equivalent to32=9.

Try It

Write the following logarithmic equations in exponential form.

  1. log10(1,000,000)=6
  2. log5(25)=2
Show Solution
  1. log10(1,000,000)=6is equivalent to106=1,000,000
  2. log5(25)=2is equivalent to52=25

 

Choose the correct exponential form for the following logarithmic equations.

 

Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the baseb, exponentx, and outputy.Then we writex=logb(y).

Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

  1. 23=8
  2. 52=25
  3. 104=110,000
Show Solution

First, identify the values ofb,y,and x.Then, write the equation in the formx=logb(y).

  1. 23=8 Here,b=2,x=3, andy=8. Therefore, the equation23=8 is equivalent tolog2(8)=3.
  2. 52=25 Here,b=5,x=2, andy=25. Therefore, the equation52=25 is equivalent tolog5(25)=2.
  3. 104=110,000 Here,b=10,x=4,andy=110,000. Therefore, the equation104=110,000 is equivalent tolog10(110,000)=4.

Try It

Write the following exponential equations in logarithmic form.

  1. 32=9
  2. 53=125
  3. 21=12
Show Solution
  1. 32=9is equivalent tolog3(9)=2
  2. 53=125is equivalent tolog5(125)=3
  3. 21=12is equivalent tolog2(12)=1

 

Select True if the equations are equivalent:

 

 

Select the correct logarithmic form of the equation:

 

 

Select True if the equations are equivalent:

 

Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, considerlog28.We ask, “To what exponent must 2 be raised in order to get 8?” Because we already know23=8, it follows thatlog28=3.

Now consider solvinglog749andlog327mentally.

  • We ask, “To what exponent must 7 be raised in order to get 49?” We know72=49.Therefore,log749=2
  • We ask, “To what exponent must 3 be raised in order to get 27?” We know33=27.Therefore,log327=3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluatelog2349mentally.

  • We ask, “To what exponent must23be raised in order to get49?” We know22=4and32=9,so(23)2=49.Therefore,log23(49)=2.

How To

Given a logarithm of the formy=logb(x), evaluate it mentally.

  1. Rewrite the argumentxas a power ofb:by=x.
  2. Use previous knowledge of powers ofbidentifyyby asking, “To what exponent shouldbbe raised in order to getx?

Solving Logarithms Mentally

Solvey=log4(64)without using a calculator.

Show Solution

First we rewrite the logarithm in exponential form:4y=64.Next, we ask, “To what exponent must 4 be raised in order to get 64?”

We know 43=64

Therefore, log4(64)=3

 

Try It

Solvey=log121(11)without using a calculator.

Show Solution

log121(11)=12(recalling that121=(121)12=11)

Evaluating the Logarithm of a Reciprocal

Evaluatey=log3(127)without using a calculator.

Show Solution

First we rewrite the logarithm in exponential form:3y=127.Next, we ask, “To what exponent must 3 be raised in order to get127?

We know33=27, but what must we do to get the reciprocal,127?Recall from working with exponents thatba=1ba.We use this information to write

33=133
 =127

Therefore,log3(127)=3.

Try It

Evaluatey=log2(132)without using a calculator.

Show Solution

log2(132)=5

Using Common Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expressionlog(x)meanslog10(x).We call a base-10 logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

Definition of the Common Logarithm

A common logarithm is a logarithm with base10.We writelog10(x)simply aslog(x).The common logarithm of a positive numberxsatisfies the following definition.

Forx>0,

y=log(x) is equivalent to 10y=x

We readlog(x)as “the logarithm with base10ofx” or “log base 10 ofx.

The logarithmyis the exponent to which10must be raised to getx.

 

How To

Given a common logarithm of the formy=log(x), evaluate it mentally.

  1. Rewrite the argumentxas a power of10:10y=x.
  2. Use previous knowledge of powers of10to identifyyby asking, “To what exponent must10be raised in order to getx?

Finding the Value of a Common Logarithm Mentally

Evaluatey=log(1000)without using a calculator.

Show Solution

First we rewrite the logarithm in exponential form:10y=1000.Next, we ask, “To what exponent must10be raised in order to get 1000?” We know

103=1000

Therefore,log(1000)=3.

 

 

Try It

Evaluatey=log(1,000,000).

Show Solution

log(1,000,000)=6

How To

Given a common logarithm with the formy=log(x), evaluate it using a calculator.

  1. Press [LOG].
  2. Enter the value given forx,followed by [ ) ].
  3. Press [ENTER].

Finding the Value of a Common Logarithm Using a Calculator

Evaluatey=log(321)to four decimal places using a calculator.

Show Solution
  • Press [LOG].
  • Enter 321, followed by [ ) ].
  • Press [ENTER].

Rounding to four decimal places,log(321)2.5065.

Analysis

Note that102=100and that103=1000.Since 321 is between 100 and 1000, we know thatlog(321)must be betweenlog(100)andlog(1000).This gives us the following:

 100<321<1000
 2<2.5065<3

Try It

Evaluatey=log(123)to four decimal places using a calculator.

Show Solution

log(123)2.0899

Rewriting and Solving a Real-World Exponential Model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation10x=500represents this situation, wherexis the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Show Solution

We begin by rewriting the exponential equation in logarithmic form.

10x=500log(500)=xUse the definition of the common log.

Next we evaluate the logarithm using a calculator:

  • Press [LOG].
  • Enter500,followed by [ ) ].
  • Press [ENTER].
  • To the nearest thousandth,log(500)2.699.

The difference in magnitudes was about2.699.

Try It

The amount of energy released from one earthquake was8,500times greater than the amount of energy released from another. The equation10x=8500represents this situation, wherexis the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Show Solution

The difference in magnitudes was about 3.929.

Using Natural Logarithms

The most frequently used base for logarithms ise.Baseelogarithms are important in calculus and some scientific applications; they are called natural logarithms. The baseelogarithm,loge(x), has its own notation,ln(x).

Most values ofln(x)can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,ln1=0.For other natural logarithms, we can use thelnkey that can be found on most scientific calculators. We can also find the natural logarithm of any power ofeusing the inverse property of logarithms.

Definition of the Natural Logarithm

A natural logarithm is a logarithm with basee. We write loge(x) simply as ln(x). The natural logarithm of a positive number x satisfies the following definition.

Forx>0,

y=ln(x) is equivalent to ey=x

We readln(x)as “the logarithm with baseeofx” or “the natural logarithm ofx.

The logarithmyis the exponent to whichemust be raised to getx.

Since the functionsy=exandy=ln(x)are inverse functions,ln(ex)=xfor allxandeln(x)=xforx>0.

 

How To

Given a natural logarithm with the formy=ln(x), evaluate it using a calculator.

  1. Press [LN].
  2. Enter the value given forx, followed by [ ) ].
  3. Press [ENTER].

Evaluating a Natural Logarithm Using a Calculator

Evaluatey=ln(500)to four decimal places using a calculator.

Show Solution
  • Press [LN].
  • Enter500,followed by [ ) ].
  • Press [ENTER].

Rounding to four decimal places,ln(500)6.2146

Try It

Evaluateln(500).

Show Solution

It is not possible to take the logarithm of a negative number in the set of real numbers.

 

Key Equations

Definition of the logarithmic function For  x>0, b>0,b1, logbx=y if and only if  by=x.
Definition of the common logarithm For x>0, y=log(x)  if and only if 10y=x.
Definition of the natural logarithm For x>0, y=ln(x)  if and only if ey=x.

Key Concepts

  • The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
  • Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm.
  • Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm.
  • Logarithmic functions with basebcan be evaluated mentally using previous knowledge of powers ofb..
  • Common logarithms can be evaluated mentally using previous knowledge of powers of10.
  • When common logarithms cannot be evaluated mentally, a calculator can be used.
  • Real-world exponential problems with base10can be rewritten as a common logarithm and then evaluated using a calculator.
  • Natural logarithms can be evaluated using a calculator.

Section Exercises

Verbal

  1. What is a baseblogarithm? Discuss the meaning by interpreting each part of the equivalent equationsby=xandlogbx=y forb>0,b1.
Show Solution

A logarithm is an exponent. Specifically, it is the exponent to which a basebis raised to produce a given value. In the expressions given, the basebhas the same value. The exponent,y, in the expressionbycan also be written as the logarithm,logbx, and the value ofxis the result of raisingbto the power ofy.

  1. How is the logarithmic functionf(x)=logbxrelated to the exponential functiong(x)=bx?What is the result of composing these two functions?
  1. How can the logarithmic equationlogbx=ybe solved forxusing the properties of exponents?
Show Solution

Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equationby=x, and then properties of exponents can be applied to solve forx.

  1. Discuss the meaning of the common logarithm. What is its relationship to a logarithm with baseb, and how does the notation differ?
  1. Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with baseb, and how does the notation differ?
Show Solution

The natural logarithm is a special case of the logarithm with basebin that the natural log always has basee.Rather than notating the natural logarithm asloge(x),the notation used isln(x).

Algebraic

For the following exercises, rewrite each equation in exponential form.

  1. log4(q)=m
  1. loga(b)=c
Show Solution

ac=b

  1. log16(y)=x
  1. logx(64)=y
Show Solution

xy=64

  1. logy(x)=11
  1. log15(a)=b
Show Solution

15b=a

  1. logy(137)=x
  1. log13(142)=a
Show Solution

13a=142

  1. log(v)=t
  1. ln(w)=n
Show Solution

en=w

For the following exercises, rewrite each equation in logarithmic form.

  1. 4x=y
  1. cd=k
Show Solution

logc(k)=d

  1. m7=n
  1. 19x=y
Show Solution

log19y=x

  1. x1013=y
  1. n4=103
Show Solution

logn(103)=4

  1. (75)m=n
  1. yx=39100
Show Solution

logy(39100)=x

  1. 10a=b
  1. ek=h
Show Solution

ln(h)=k

For the following exercises, solve forxby converting the logarithmic equation to exponential form.

  1. log3(x)=2
  1. log2(x)=3
Show Solution

x=23=18

  1. log5(x)=2
  1. log3(x)=3
Show Solution

x=33=27

  1. log2(x)=6
  1. log9(x)=12
Show Solution

x=912=3

  1. log18(x)=2
  1. log6(x)=3
Show Solution

x=63=1216

  1. log(x)=3
  1. ln(x)=2
Show Solution

x=e2

For the following exercises, use the definition of common and natural logarithms to simplify.

  1. log(1008)
  1. 10log(32)
Show Solution

32

  1. 2log(.0001)
  1. eln(1.06)
Show Solution

1.06

  1. ln(e5.03)
  1. eln(10.125)+4
Show Solution

14.125

Numeric

For the following exercises, evaluate the baseblogarithmic expression without using a calculator.

  1. log3(127)
  1. log6(6)
Show Solution

12

  1. log2(18)+4
  1. 6log8(4)
Show Solution

4

For the following exercises, evaluate the common logarithmic expression without using a calculator.

  1. log(10,000)
  1. log(0.001)
Show Solution

3

  1. log(1)+7
  1. 2log(1003)
Show Solution

12

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

  1. ln(e13)
  1. ln(1)
Show Solution

0

  1. ln(e0.225)3
  1. 25ln(e25)
Show Solution

10

Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

  1. log(0.04)
  1. ln(15)
Show Solution

2.708

  1. ln(45)
  1. log(2)
Show Solution

0.151

  1. ln(2)

Extensions

  1. Isx=0in the domain of the functionf(x)=log(x)?If so, what is the value of the function whenx=0?Verify the result.
Show Solution

No, the function has no defined value forx=0.To verify, supposex=0is in the domain of the functionf(x)=log(x).Then there is some numbernsuch thatn=log(0).Rewriting as an exponential equation gives:10n=0, which is impossible since no such real numbernexists. Therefore,x=0is not the domain of the functionf(x)=log(x).

  1. Isf(x)=0in the range of the functionf(x)=log(x)?If so, for what value ofx?Verify the result.
  1. Is there a numberxsuch thatlnx=2?If so, what is that number? Verify the result.
Show Solution

Yes. Suppose there exists a real numberxsuch thatlnx=2.Rewriting as an exponential equation givesx=e2, which is a real number. To verify, letx=e2.Then, by definition,ln(x)=ln(e2)=2.

  1. Is the following true:log3(27)log4(164)=1?Verify the result.
  1. Is the following true:ln(e1.725)ln(1)=1.725?Verify the result.
Show Solution

No;ln(1)=0, soln(e1.725)ln(1)is undefined.

Real-World Applications

  1. The exposure indexEIfor a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equationEI=log2(f2t), wherefis the “f-stop” setting on the camera, and t is the exposure time in seconds. Suppose the f-stop setting is8and the desired exposure time is2seconds. What will the resulting exposure index be?
  1. Refer to the previous exercise. Suppose the light meter on a camera indicates anEIof2, and the desired exposure time is 16 seconds. What should the f-stop setting be?
Show Solution

2

  1. The intensity levels I of two earthquakes measured on a seismograph can be compared by the formulalogI1I2=M1M2whereMis the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

Footnotes

4http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.

5http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013.

6http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013.

7http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.

8http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014.

Glossary

common logarithm
the exponent to which 10 must be raised to getx;log10(x) is written simply aslog(x).
logarithm
the exponent to whichbmust be raised to getx;writteny=logb(x)
natural logarithm
the exponent to which the numberemust be raised to getx;loge(x)is written asln(x).

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